Problem: The mean of the increasing ordered set of numbers, $\{6, x, 22\}$, is equal to the median. What is the value of $x$?
Solution: The mean of the set is the sum of all the numbers divided by the number of numbers (which is 3), so the mean is $\frac{6+x+22}{3}$. The median of the set is the number in the middle when the numbers are written in increasing order, so the median of this set is $x.$ Thus, we have \[\frac{6+x+22}{3} = x.\] Multiplying both sides by 3 yields $6+x+22 = 3x$, which simplifies to $28 = 2x$ or $x=\boxed{14}$.